M07_MCDO8122_01_ISM_C07

M07_MCDO8122_01_ISM_C07 - Chapter 7 Interest Rate Forwards...

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Chapter 7 Interest Rate Forwards and Futures n Question 7.1 We can use (7.1) and solve for the effective annual yield as follows: 1 1/ 1 (0, ) [1 (0, )] [1 (0, )] (0, ) (0, ) (0, ) 1 n n n P n r n r n P n r n P n - - = + + = = - We can determine the continuous rate for maturity T , (0, ), cc r T from the zero-coupon bond prices; specifically, (0, ) (0, ) 1 1 (0, ) ln [1/ (0, )] 1 ln [ (0, )] cc r T T cc e P T r T P T T P T T = = = - For the forward rates and par coupon rates, we can use Equations (7.3) and (7.6). We obtain the following yields and prices: Maturity Zero-Coupon Bond Yield Zero Coupon Bond Price One-Year Implied Forward Rate Par Coupon Cont. Comp. Zero Yield 1 0.04000 0.96154 0.04000 0.04000 0.03922 2 0.04500 0.91573 0.05003 0.04489 0.04402 3 0.04500 0.87630 0.04500 0.04492 0.04402 4 0.05000 0.82270 0.06515 0.04958 0.04879 5 0.05200 0.77611 0.06003 0.05144 0.05069 Note that we have chosen to put the one year implied forward rate from year n to n + 1 in maturity n + 1 row. We demonstrate the calculations for a maturity of four years. The zero-coupon bond yield is derived from Equation (7.1) and the above result: 1/3 1/3 (0, 4) (0, 4) 1 0.8227 1 0.05 r P - - = - = - =
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80 McDonald • Fundamentals of Derivatives Markets The one year implied forward rate from Year 3 to Year 4 is determined by Equation (7.3): 0 (0, 3) 0.8763 (3, 4) 1 1 0.06515 (0, 4) 0.8227 P r P - = - = The par coupon rate for a four year maturity uses Equation (7.6) with 0, t = 4, T = and four yearly coupons: 1 (0, 4) (0,1) (0, 2) (0, 3) (0, 4) 1 0.8227 0.96154 0.91573 0.8763 0.8227 0.04958 P c P P P P - = + + + - = + + + = The four year continuous yield is: 4 ln(0.8227) 0.04879 4 r - = = n Question 7.2 The coupon bond pays a coupon of $60 each year plus the principal of $1,000 after five years. We have cash flows of 60, 60, 60, 60,1060. To obtain the price of the coupon bond, we multiply each cash flow by the zero-coupon bond price of that year. Specifically, Bond Price 60 ( (0,1) (0, 2) (0, 3) (0, 4) (0, 5)) 1000 (0, 5) 60 (0.96154 0.91573 0.87630 0.8227 0.77611) 1000 0.77611 60 4.3524 776.11 261.14 776.11 1037.25 P P P P P P = × + + + + + × = × + + + + + × = × + = + = This yields a bond price of $1,037.25. n Question 7.3 This is a straightforward application of Equations (7.1), (7.3), and (7.6). We also need the continuous rate calculation (derived in Question 7.1): ln(1/ (0, )) ln( (0, )) (0, ) cc P T P T r T T T - = = Maturity Zero-Coupon Bond Yield Zero Coupon Bond Price One-Year Implied Forward Rate Par Coupon Cont. Comp. Zero Yield 1 0.03000 0.97087 0.03000 0.03000 0.02956 2 0.03500 0.93351 0.04002 0.03491 0.03440 3 0.04000 0.88900 0.05007 0.03974 0.03922 4 0.04500 0.83856 0.06014 0.04445 0.04402 5 0.05000 0.78353 0.07024 0.04903 0.04879
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Chapter 7 Interest Rate Forwards and Futures 81 n Question 7.4 To solve this problem, we first generate the zero-coupon bond prices from the implied one-year forward rates that are given. All other rates follow as in Question 7.1.
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This note was uploaded on 02/27/2010 for the course FIN 311 taught by Professor Haan during the Spring '10 term at St. Josephs NY.

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M07_MCDO8122_01_ISM_C07 - Chapter 7 Interest Rate Forwards...

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